A090319 Fifth column (k=4) of triangle A084938.

1, 4, 14, 52, 217, 1040, 5768, 36992, 272584, 2285184, 21550656, 226071744, 2611146384, 32911082496, 449243785728, 6598780563456, 103734755882496, 1737181702840320, 30866291090657280, 579859321408266240

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..150

Formula

a(n) = Sum_{k=0..n} A090595(k)*(n-k)!.

a(n) = Sum_{a+b+c+d = n} a!*b!*c!*d!.

a(n) = Sum_{k=0..n} A003149(k)*A003149(n-k).

G.f.: (Sum_{k>=0} k!*x^k)^4.

From G. C. Greubel, Dec 29 2019: (Start)

a(n) = (n+3)!*Sum_{k=0..n} Sum_{m=0..k} Sum_{j=0..m} Beta(k+3, n-k+1)*Beta(m+2, k-m+1)*Beta(j+1, m-j+1), where Beta(x,y) is the Beta function.

a(n) = Sum_{k=0..n} Sum_{m=0..k} Sum_{j=0..m} n!/(binomial(n,k) * binomial(k,m) * binomial(m,j)). (End)

Maple

seq( (n+3)!*add(add(add( Beta(k+3, n-k+1)*Beta(m+2, k-m+1)*Beta(j+1, m-j+1), j=0..m), m=0..k), k=0..n), n=0..20); # G. C. Greubel, Dec 29 2019

Mathematica

Table[(n+3)!*Sum[Beta[k+3, n-k+1]*Beta[m+2, k-m+1]*Beta[j+1, m-j+1], {k, 0, n}, {m, 0, k}, {j, 0, m}], {n, 0, 20}] (* G. C. Greubel, Dec 29 2019 *)

Prog

(PARI) vector(21, n, my(b=binomial); sum(k=0, n-1, sum(m=0, k, sum(j=0, m, (n-1)!/(b(n-1, k)*b(k, m)*b(m, j)) )))) \\ G. C. Greubel, Dec 29 2019
(Magma) F:=Factorial; B:=Binomial; [ (&+[(&+[(&+[F(n)/(B(n, k)*B(k, m)*B(m, j)): j in [0..m]]): m in [0..k]]): k in [0..n]]): n in [0..20]]; // G. C. Greubel, Dec 29 2019
(Sage) b=binomial; [sum(sum(sum(factorial(n)/(b(n, k)*b(k, m)*b(m, j)) for j in (0..m)) for m in (0..k)) for k in (0..n)) for n in (0..20)] # G. C. Greubel, Dec 29 2019
(GAP) B:=Binomial;; List([0..20], n-> Sum([0..n], k-> Sum([0..k], m-> Sum([0..m], j-> Factorial(n)/(B(n, k)*B(k, m)*B(m, j)) )))); # G. C. Greubel, Dec 29 2019

Crossrefs

Cf. A084938.

Columns, for k = 0, 1, 2, 3 : A000007, A000142, A003149, A090595.

Sequence in context: A308023 A149489 A125783 * A295518 A047118 A017948

Adjacent sequences: A090316 A090317 A090318 * A090320 A090321 A090322

Keyword

easy,nonn

Author

Philippe Deléham, Feb 05 2004

Status

approved